Library / Control engineering / CTL-07
Control engineering
Steady-state error to a step input
Final tracking error of a stable unity-feedback loop for a step reference of size R.
- Ref
- CTL-07
- Rev
- v1
- Field
- Control engineering
- Result
- ess
For a stable unity-feedback loop L(s), a step of size R produces a final error R/(1 + Kpos), where Kpos = lims→0 L(s) is the zero-frequency loop gain. If the loop contains an integrator, Kpos is infinite and the step error is zero. Kpos is a loop-gain constant and has nothing to do with the proportional gain of a PID controller.
Final tracking error of a stable unity-feedback loop for a step reference of size R. Kpos is the loop gain at zero frequency; a loop with an integrator has infinite Kpos and zero step error.
Calculator
- Error as a percentage of the step10 %
- Final output value, R − ess0.9
- Kpos needed for a 1 % error99
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With Kpos = 9 the loop holds the output to within 10 % of the step. A 1 % error needs Kpos = 99, and the error goes to zero only if the loop has an integrator.
- Doubling Kpos to 18 reduces the error from 0.1 to 0.05263: it falls by less than half because of the 1 in the denominator.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Size of the reference step | 0.5 to 10 | 0.05 | 1 | 0.95 | +10% | +10 % |
| Kpos Position error constant, L(0) | 1 to 100 | 0.009901 | 0.5 | 0.4901 | +10% | −8.26 % |
Sweep of Position error constant, L(0)
| Position error constant, L(0) | Steady-state error |
|---|---|
| 1 | 0.5 |
| 13.38 | 0.06957 |
| 25.75 | 0.03738 |
| 38.13 | 0.02556 |
| 50.5 | 0.01942 |
| 62.88 | 0.01566 |
| 75.25 | 0.01311 |
| 87.63 | 0.01128 |
| 100 | 0.009901 |
Sweep one variable
Variables and how each one contributes
Steady-state error
What the equation gives you. Everything else feeds it.
Size of the reference step
The size of the reference step. Directly proportional: a bigger step leaves a proportionally bigger final error. Raising it by 10 % (to 1.1) moves the result by +10 %, from 0.1 to 0.11.
Position error constant, L(0)
The position error constant, the value of the open-loop transfer function at zero frequency. A higher loop gain at zero frequency shrinks the error, with diminishing returns as 1 + Kpos grows. Raising it by 10 % (to 9.9) moves the result by −8.26 %, from 0.1 to 0.09174.
Worked example with the values above
- Denominator: 1 + Kpos = 1 + 9 = 10.
- Error: ess = R / 10 = 1 / 10.
- ess = 0.1, so the output settles at 0.9.
Assumptions and limits
- A stable closed loop with unity negative feedback and no feedforward, so that the final value theorem can be used.
- A constant (step) reference and no disturbance. Disturbances add their own steady error.
Common mistakes
- Using it for an unstable loop. The final value theorem needs every pole of sE(s) to lie in the left half-plane.
- Confusing Kpos with the proportional gain Kp of a PID controller. Kpos includes the plant gain.
Statistics and uncertainty
The error is R/(1 + Kpos), so its relative sensitivity to Kpos is Kpos/(1 + Kpos), which is 0.9 at the defaults. A 10 % error in the gain therefore gives nearly a 10 % error in ess. A plant whose zero-frequency gain drifts with temperature or load will give a drifting steady error unless integral action is used.
Where it is used
- Deciding whether integral action is needed for a set-point to be held exactly.
- Choosing a loop gain to meet a steady-state accuracy specification.
Maths you need
- Final value theorem: lim t→∞ e(t) = lim s→0 sE(s)
- Open-loop transfer function: L = C G, the controller times the plant